DISCRETE EVENT CONTROL OF TIME VARYING PLANTS

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1 DISCRETE EVENT CONTROL OF TIME VARYING PLANTS Ernesto KOFMAN Laboratorio de Sistemas Dinmicos. FCEIA UNR CONICET. Riobamba 245 bis (2) Rosario Argentina Abstract This paper studies the use of Quantized State Control (QSC) in Time Varying (TV) plants. Making use of a Lyapunov analysis, the stability properties of Time Invariant QSC are extended to the non stationary case. Then, based on the resulting stability theorem, a design algorithm is developed. Finally, the use of this algorithm which allows the design of QSC controllers according to stability and convergence speed features is shown with the design and the simulation of an illustrative example. Keywords Digital Control, Nonlinear Control, Discrete Event Systems. I. INTRODUCTION Quantized State Control (Kofman, 23) is a methodology which allows the digital implementation of previously designed continuous controllers based on the quantization of their state variables. The resulting digital controller which can be described by a discrete event system is completed with an asynchronous sampling scheme (Sayiner et al., 1993) so that, ignoring the temporal errors introduced by the clock of the digital device, no time discretization is performed. The use of QSC instead of classic discrete time digital controllers permits to conserve the region of attraction in nonlinear systems improving also the dynamic response and reducing the computational costs as well as the information transmitted between controllers and plants. In the LTI case, digital QSC controllers also ensure that the resulting trayectories do not differ from the ones obtained with the ideal continuous controllers in more than a bound which can be calculated with a closed formula (Kofman, 22). Since QSC takes into account the presence of quantization, it does not guarantee asymptotic stability but ultimately boundedness of the solutions (Khalil, 1996). This fact, which seems to be a disadvantage, is common to all the quantization based approaches (Delchamps, 199; Brockett and Liberzon, 2; Elias and Mitter, 21). Moreover, in any sampled data control system where the effects of A/D and D/A converters are taken into account the theory can only talk about ultimate bounds (Miller et al., 1989; Hou et al., 1997). The original definitions of QSC and the mentioned properties were established under the assumption of a Time Invariant (TI) plant with a TI controller. Although the restriction on the controller cannot be avoided 1, there is no reason to consider only TI plants. Taking into account the last remark, this work attempts to extend the properties and design algorithms of TI QSC to the Time Varying case. The paper is organized as follows: After recalling the principles and properties of QSC (Section II), the main results of QSC with TV plants a stability Theorem and a design algorithm are presented in Section III. Finally, these results are illustrated with the design and simulation of a simple example. II. QUANTIZED STATE CONTROL QSC is a digital control scheme in which a continuous plant is controlled by a discrete event system obtained with the quantization of a continuous controller. A. Quantized State Systems Consider the State Equation System (SES) given by: ẋ(t) = f(x(t),u(t)) (1) y(t) = g(x(t),u(t)) Related to this system, an associated Quantized State System (QSS) (Kofman and Junco, 21) is given by: ẋ(t) = f(q(t),u(t)) (2) y(t) = g(q(t),u(t)) In this system, q(t) (quantized variables) and x(t) (state variables) are related (componentwise) by hysteretic quantization functions, which are defined as follows: Definition 1. Hysteretic Quantization Function. Let Q = Q,Q 1,..., Q r } be a set of real numbers where Q k 1 <Q k with 1 k r. Let Ω be the set of piecewise continuous real valued trajectories and let x i Ω be a continuous trajectory. Let b :Ω Ω be 1 In the general case it is not possible to obtain a discrete event description of a quantized time varying system.

2 a mapping and let q i = b(x i ) where the trajectory q i satisfies: Q m if t = t Q k+1 if x i (t) =Q k+1 q q i (t) = i (t )=Q k k<r (3) Q k 1 if x i (t) =Q k ε q i (t )=Q k k> q i (t ) otherwise and if x i (t ) <Q m = r if x i (t ) Q r j if Q j x(t ) <Q j+1 Then, the map b is a hysteretic quantization function. The discrete values Q k are called quantization levels and the distance Q k+1 Q k is defined as the quantum, which is usually constant. The width of the hysteresis window is ε and, as it was shown in (Kofman et al., 21), it should be taken equal to the quantum to improve the accuracy. The values Q and Q r are the lower and upper saturation bounds. A fundamental property of a Quantization Function with hysteresis when t t and Q x i (t) Q r is given by the following inequality q i (t) x i (t) max 1 i r (Q i Q i 1,ε) (4) The role of the quantization functions in (2) is to convert the state trajectories (x(t)) into piecewise constant ones (q(t)). In that way, provided that u(t) is piecewise constant, it results that the derivatives ẋ(t) are also piecewise constant and then the state trajectories are piecewise linear. As a consequence of these properties of the trajectory forms, QSS can be exactly represented by discrete event models within the DEVS formalism framework (Zeigler et al., 2). The DEVS model related to a generic QSS and the proof of the mentioned properties can be found in (Kofman and Junco, 21). The possibility of representing QSS by DEVS models and the fact that DEVS models can be simulated in real time by digital devices 2 (Zeigler and Kim, 1993) suggested the use of QSS as digital controllers and the definition of QSC. B. QSC Definition Consider the Continuous Control System (CCS) consisting of plant and controller, Eqs.(5) and (6) respectively, and their (ideal) interconnection, Eq. (7). ẋp (t) = f p (x p (t),u p (t),t) (5) y p (t) = g p (x p (t),t) ẋc (t) = f c (x c (t),u c (t),u r (t)) (6) y c (t) = g c (x c (t),u c (t),u r (t)) 2 DEVS representation of QSS is exact. However, real time simulation of DEVS has errors related to temporal resolution and round-off in digital devices u p (t) =y c (t), u c (t) =y p (t) (7) It is being considered here that the plant could be time varying. When it comes to the controller, it is assumed that it is stationary and it has an input reference u r (t). Definition 2. Quantized State Controller. A QSS associated to a continuous controller (6) is called Quantized State Controller (QSC controller). Definition 3. QSC System. A QSC system is a control scheme composed by a continuous plant and a QSC controller connected through asynchronous A/D and D/A converters. D/A y c g c u p x p q c f p Controller Plant f c g p q c u c y p A/D Figure 1: Block Diagram of the QSC system Figure 1 shows a block diagram representation of a QSC system. Despite its state equation like representation, the controller is in fact a DEVS model. The QSC implementation of the controller transforms (6) into the new set of equations: ẋc (t) = f c (q c (t),u c (t),u r (t)) (8) y c (t) = g c (q c (t),u c (t),u r (t)) The asynchronous sampling scheme (Sayiner et al., 1993) implies that the A/D conversions are performed only when the analog input and the digital output of the converters differ in a quantity corresponding to one quantization interval. Then, they can be seen as quantization functions with hysteresis where the quantization intervals and the hysteresis windows have the same size. Similarly, D/A converters can be represented by quantization functions (but without hysteresis). Thus, the presence of the asynchronous converters transforms (7) into: u r u p (t) =y cq (t), u c (t) =y pq (t) (9) where variables y cq (t) andy pq (t) are quantized versions of the plant and the controller output variables

3 C. QSC Properties The CCS closed loop equations can be derived from Eqs.(5) (7) arriving to ẋp = f p (x p,g c (x c,g p (x p,t),u r ),t) (1) ẋ c = f c (x c,g p (x p,t),u r ) Let us define x c (t) = q c (t) x c (t) (11a) y p (t) = y pq (t) y p (t) (11b) y c (t) = y cq (t) y c (t) (11c) Thus, from these definitions and Eqs.(5), (8) and (9), the QSC closed loop equations can be written as: ẋ p = f p (x p,g c (x c + x c,g p (x p,t)+ + y p,u r )+ y c,t) ẋ c = f c (x c + x c,g p (x p,t)+ y p,u r ) (12) Then, the QSC system (12) can be seen as a perturbed version of the original CCS (1). From the property of the quantization functions provided that the variables x c, y p and y c do not reach the corresponding saturation bounds 3 it also results that the perturbation terms are bounded by (4). Thus, the properties of the QSC implementation of a CCS can be studied by looking at the effects of bounded perturbations in the original closed loop system. Based on this remark, the stability theorem of TI QSC (Kofman, 23) and the stability and error bound estimation of LTI QSC (Kofman, 22) were proven. The stability theorem of TI QSC tells that, given a CCS in which the origin is an asymptotically stable equilibrium point, an appropriate quantization can be found so that the resulting QSC controller ensures ultimately boundedness of the solutions (for any given ultimate bound) conserving also the estimate region of attraction. That appropriate quantization can be found according to desired ultimate bound following an algorithm which makes use of a Lyapunovfunction of the original CCS. On the other hand, the theorem for LTI QSC tells that the QSC implementation of an asymptotically stable LTI CCS is always globally and ultimately bounded. The ultimate bound can be estimated with a closed formula. Moreover, the trajectories of the CCS and the resulting QSC never differ from each other in more than the mentioned bound. Thus, the quantization choice which completes the QSC design can be made based on these properties, according to the desired ultimate bounds. III. TIME VARYING QSC The stability theorem and the corresponding design algorithm were based on a stationary Lyapunovanalysis. This section introduces the main contributions 3 It will be considered that the non saturation region is enough large so that the trajectories do not leave it. of this work by extending those results to the time varying case. A. Stability of TV QSC When the reference trajectory u r (t) is zero (or constant), the QSC system (12) can be rewritten as ẋ = f(x + x, y, t) (13) where x, [x p,x c ] T, x, [, x c ] T, y, [ y p, y c ] T,andf, [f p,f c ] T. With these definitions, the CCS (1) becomes ẋ = f(x,,t), f(x, t) (14) Then, the following theorem which relates the stability properties of (14) and (13) can be stated: Theorem 1. Let the origin be an asymptotically stable equilibrium point of the closed loop CCS (14). Assume that function f is continuous and a continuously differentiable Lyapunov function V (x, t) is known with W 1 (x) V (x, t) W 2 (x) (15) V f(x, t)+ V t W 3(x) (16) t, x D being D a compact set which contains the origin and W i are continuous positive definite functions in D. Let Ω 2a = x W 2 (x) a} with a being an arbitrary positive constant which is enough small so that x W 1 (x) a} is a closed region inside D. Let Ω 1b = x W 1 (x) b} being b an arbitrary positive constant (b <a) enough small so that Ω 1b Ω 2a. Then, a quantization can be found so that the QSC system trajectories starting in Ω 2a finish inside Ω 1b, reaching this region in finite time. Proof. The derivative of V (x, t) along the solutions of the QSC system (13) is V (x, t) = V = V + V f(x + x, y, t)+ V t f(x,,t)+ V t + Then, using (16) it results that (f(x + x, y, t) f(x,,t)) V (x, t) W 3 (x)+ V (f(x+ x, y, t) f(x,,t)) Consider the sets Ω 2b = x W 2 (x) b} and Ω 1a = x W 1 (x) a}. From the hypotesis made about a and b, it results that Ω 2b Ω 1b Ω 2a Ω 1a D (17) Since W 3 (x) is positive definite in D, it is positive in Ω 1,2, Ω 1a Ω 2b. Moreover, it exists a positive constant s where s, min x Ω 1,2 W 3 (x) (18)

4 Let us define the following function α(x, x, y, t), W 3 (x)+ V (19) (f(x + x, y, t) f(x,,t)) The continuity in functions W 3 and f and the fact that V is continuously differentiable implies that α is continuous. From the definition of α and Eq.(18) it results that α(x,,,t) s x Ω 1,2, t (2) Let α M be the function defined by α M ( x, y), sup x Ω 1,2,t (α(x, x, y, t)) (21) It can be easily seen that α M is continuous and α M (, ) s. Then, for any positive number s 1 <s a positive constant r can be found so that the condition ( x, y) r (22a) implies that α M ( x, y) s 1 (22b) and then it results that V (x, t) α(x, x, y, t) α M ( x, y) s 1 (23) in Ω 1,2. From here to the end, the proof follows Theorem 5.3 of (Khalil, 1996). Let Ω a,t = x V (x, t) a} and Ω b,t = x V (x, t) b} be time variable sets. From (15) it results that Ω 2b Ω b,t Ω 1b Ω 2a Ω a,t Ω 1a D (24) The border of Ω a,t is inside Ω 1,2, where V (x, t) is negative. It means that the trajectories of the QSC system (13) cannot abandon Ω a,t. Then, any trajectory starting in Ω 2a cannot abandon Ω 1a. The border of Ω b,t is also inside Ω 1,2. Then the trajectories cannot abandon this time variable set. To complete the proof, we need to ensure that the trajectories initiated in Ω 2a Ω a,t reach Ω b,t in a finite time. Let φ(t) be a solution of (13) starting in Ω a,t (i.e. V (φ(), ) a) and let us suppose that V (φ(t),t) >b t (25) then we have V (φ(t),t) s 1 and after t 1, a b (26) s 1 it results that V (φ(t 1 ),t 1 ) b which yields a contradiction. Then, the region Ω b,t must be reached before the finite time t 1. Since Ω b,t Ω 1b the trajectory also reaches the set before that time. Ω 1b Equation (22a) gives the maximum perturbation allowed to ensure the achievement of the proposed goal (i.e. region of attraction Ω 2a and ultimate bound in Ω 1b ). Since the maximum perturbation in each variable is given by the corresponding quantum, this equation should be used to choose the quantum at the different controller state variables and converters completing in that way the QSC design. Observe that Ω 2a is also the estimation of the region of attraction of the CCS using the Lyapunovfunction V. Then, a QSC implementation can be found so that it conserves the estimated region of attraction. B. Design Algorithm for TV QSC The design of a QSC contoller can be divided in two steps. The first one is the design of the continuous controller, which can be done following any technique. The second step is the choice of the quantization at each variable. The use of a very small quantum yields solutions which are very closed to the trajectories of the CCS. This is due to the property of convergence which tells that under locally Lipchitz conditions on the functions the solutions of the QSC system go to the solutions of the CCS when the quantization go to zero (Kofman, 23). In that way, and also according to Theorem 1, the ultimate bound can be reduced to arbitrary small values. However, the use of a small quantum increases the number of events at the controller and the digital device can fail in its attempt to give the correct output values at the required time. Therefore, there is always a trade off between accuracy and practical considerations related to the computational costs. Then the idea is to exploit Theorem 1 in order to choose the quantization according to some essential features (region of attraction and ultimate bound). In that way, the quantization adopted will be just as small as necessary to ensure those properties and provided that the CCS is not too fast the digital device will be able to correctly implement the resulting QSC controller. The translation of these ideas into a design algorithm for QSC can be written as follows: 1. Design a continuous controller and calculate the Lyapunovfunction V (x, t) and the functions W i (x) according to (15) (16) for the closed loop CCS. 2. Choose the QSC region of attraction Ω 2a and the ultimate region Ω 1b together with the constants a and b. 3. Obtain the perturbed closed loop function f according to (13). 4. Obtain function α according to (19) and α M according to (21).

5 5. Calculate constant s according to (18) and choose the positive constant s 1 <s. If the goal is just to ensure ultimately boundedness s 1 should be very small. Otherwise, if the speed of convergence is also important, it can be chosen taking into account (26). 6. Compute the value of r according to (22). 7. Choose the quantization at the controller state variables and converters so that (22a) is satisfied. It can be easily seen that this procedure leads to a QSC controller which ensures region of attraction Ω 2a and ultimate region Ω 1b. IV. EXAMPLES AND RESULTS The unstable time varying plant ẋp = x p (1 + sin t +cost)+u p y p = (2 + cos t) x p can be stabilized by the controller ẋc = x c + u c y c = x c u c (27) The resulting closed loop system can be written as ẋp = (1 sin t) x p x c ẋ c = (2 + cos t) x p x c Here, the Lyapunovcandidate verifies (15) with V (x p,x c,t)=x 2 p x2 c x2 p cos t W 1 (x p,x c )= 1 2 x2 p x2 c (28) The ultimate bound Ω 1b will be chosen with b =.5. Then, taking into account (28) it results that Ω 1b = x R 2 x 1}. The perturbed equations (13) can be written as ẋp = (1 sin t) x p x c x c + y c y p ẋ c = (2 + cos t) x p x c x c + y p and then, from (19) function α(x, x, y, t) results α = x 2 c 1 2 x2 p + x p (2 + cos t) ( x c + y c y p )+x c ( x c + y p ) Although the maximum α M in (21) cannot be easily obtained, it can be bounded using the fact that x > 1/3 inω 1,2. Then, after some calculations it results that α M ( ( x c + y c + y p ) ( x c + y p ) 2) 1/2 Thus, taking the quantum equal to.18 in the controller and converters we can ensure that α M.93 in Ω 1,2, which implies that the trajectories finish inside Ω 1b in finite time, with a minimum speed s 1 =.93. Figs.2 5 show the simulation results for an initial condition x p =5,x c = The orbital derivative is W 2 (x p,x c )= 3 2 x2 p x2 c V = (2 + cos t) (sin t 1) x 2 p x 2 c 1 2 x2 p sin t which satisfies (16) with W 3 = 1 2 x2 p x 2 c Then, the closed loop CCS is asymptotically stable and the algorithm resulting from Theorem 1 can be used to design the QSC controller. The first step for the QSC design consists in choosing the region of attraction and the ultimate bound. Since inequalities (15) (16) stand in R 2 (i.e. the CCS stability is global) it is not necessary to restrict the region of attraction except for chosing the saturation values. In this case, the choice of Ω 2a does not affect the calculations Figure 2: x p vs. t Figure 3: Final oscillations in x p vs. t

6 5 4 3 that case, the result was very conservative. If the geometrical analysis for LTI systems of (Kofman, 22) were extended for LPV plants, less conservative results might be obtained. 2 REFERENCES Figure 4: x p vs. x c Figure 5: Final oscillations in x p vs. x c Fig.5 also shows that the design was very conservative. The ultimate bound observed in the simulation is less than.3 (in norm 2) which is more than 3 times smaller than the estimated. There are two reasons which can explain this. The first one is that Lyapunovanalysis often leads to conservative estimations of the ultimate bounds. The second reason is that in this case the D/A converter does not introduce any perturbation since it is exactly matched with the quantizer of x c and the A/D converter. V. CONCLUSIONS This work studied the properties of QSC in time varying systems showing that the main features and advantages of the methodology are also satisfied in these cases. Theorem 1 proved that the QSC control of time varying plants allows to ensure ultimately boundedness conserving the region of attraction. A design algorithm derived from this theorem was also introduced and its use was illustrated with an example. When it comes to future work, it should be considered that the main theoretical properties were already proven for general nonlinear time varying systems and for LTI systems. There is an intermediate case which should be taken into account which corresponds to Linear Parameter Varying (LPV) plants. The example given in fact corresponds to that category and although a Lyapunovanalysis could be done for Brockett, R. and D. Liberzon, Quantized feedback stabilization of linear systems, IEEE Trans. Automat. Contr. 45, (2). Delchamps, D., Stabilizing a linear system with quantized state feedback, IEEE Trans. Automat. Contr. 35, (199). Elias, N. and S. Mitter, Stabilization of Linear Systems with Limited Information, IEEE Trans. Automat. Contr. 46(9), (21). Hou, L., A. Michel, and H. Ye, Some qualitative properties of sampled-data control systems, IEEE Trans. Automat. Contr. 42, (1997). Khalil, H., Nonlinear Systems, Prentice-Hall, New Jersey, 2nd edition (1996). Kofman, E., Quantized-State Control of Linear Systems, Proceedings of AADECA 22, Buenos Aires, Argentina (22). Kofman, E., Quantized-State Control. A Method for Discrete Event Control of Continuous Systems., Latin American Applied Research 33(4), (23). Kofman, E. and S. Junco, Quantized State Systems. A DEVS Approach for Continuous System Simulation, Transactions of SCS 18(3), (21). Kofman, E., J. Lee, and B. Zeigler, DEVS Representation of Differential Equation Systems. Review of Recent Advances, Proceedings of ESS 1 (21). Miller, R., A. Michel, and J. Farrel, Quantizer effects on steady-state error specifications of digital feedback control systems, IEEE Trans. Automat. Contr. 34, (1989). Sayiner, N., H. Sorensen, and T. Viswanathan, A non-uniform sampling technique for A/D conversion, Proceedings of ISCAS 93 2, (1993). Zeigler, B. and J. Kim, Extending the DEVS- Scheme knowledge-based simulation environment for real time event-based control, IEEE Trans. on Robotics and Automation 3(9), (1993). Zeigler, B., T. Kim, and H. Praehofer, Theory of Modeling and Simulation. Second edition, Academic Press, New York (2).